Bring water and salt to a boil over medium-high. Slowly stir in polenta and reduce heat to medium. Continue to cook, stirring constantly, until the polenta is very thick (about 10 minutes). Add in herbs and cheese. Spray a glass dish or rimmed baking sheet with cooking oil and spread polenta ½ an inch thick. Let cool in fridge for at least 30 minutes or freezer for 15 minutes.
Meanwhile, make dressing: Mince garlic and anchovies, blending into a paste with the side of your knife. (If using anchovy paste, mix garlic into paste.) Whisk egg yolks in medium bowl. Add garlic-anchovy paste, mustard and lemon juice and whisk until incorporated. Pour olive oil into bowl in thin stream while whisking continuously, until the oil is gone and the dressing is emulsified. Add salt and pepper to taste and set aside.
Preheat oven to 450ºF. Once polenta has cooled, cut it into ½-inch squares and place on oiled baking sheet, making sure croutons are spread out. Bake 25-30 minutes, turning croutons halfway through to ensure they brown evenly. Set aside to cool.
Assemble chicken, lettuce and tomatoes in large salad bowl or platter. Top with Parmesan and croutons and serve with dressing.
Reserve leftover croutons and dressing separately for later servings. Store both in fridge.